Proof Theory of Martin-L{\"o}f Type Theory
The goal of Hilbert's was to show the consistency of strong axiomatizations of
mathematics by finitary methods. By G{\"o}del's incompleteness theorem we know
that this is not possible. One solution is taken in ordinal theoretic
proof theory, where the consistency of theories is shown by
transfinite induction over a suitable ordinal notation system.
While this provides some evidence of the consistency for medium sized theories,
for strong theories a direct insight into the well-foundedness of the
ordinal notation system seems at the moment
to be very difficult.
One solution is to replace finitary methods by a theory
for which we have some evidence into its consistency. The theory which
has been developed most in order to serve for such a purpose is Martin-L{\"o}f
Type Theory.
A revised Hilbertian program then can be carried out in two steps: First
the ordinal analysis of a theory as described before is carried out.
Then the well-foundedness of the ordinal notation system used is shown
in a suitable extension of Martin-L{\"o}f Type Theory.
This requires the development of strong extensions of Martin-L{\"o}f type theory
such that we have some evidence of their consistency.
The application of this approach is the development of proof theoretically
strong data structures, which can then be used in general programming.
In this talk we describe some steps in this program:
Martin-L{\"o}f Type Theory with W-type and one universe which
proves consistency of Kripke Platek set theory with one recursively inaccessible;
the Mahlo universe, which proves consistency of Kripke Platek set theory
with one recursively Mahlo ordinal; and an extension of
type theory by the principle of $\Pi_3$-reflection.
We will discuss the controversy about the Mahlo universe, and
how one can overcome this problem when switching to a framework
of explicit mathematics. This shows that a framework involving
partial functions might be more suitable for proving the
consistency of proof theoretically strong theories.